Accurate lineshape spectroscopy and the Boltzmann constant

Spectroscopy has an illustrious history delivering serendipitous discoveries and providing a stringent testbed for new physical predictions, including applications from trace materials detection, to understanding the atmospheres of stars and planets, and even constraining cosmological models. Reaching fundamental-noise limits permits optimal extraction of spectroscopic information from an absorption measurement. Here, we demonstrate a quantum-limited spectrometer that delivers high-precision measurements of the absorption lineshape. These measurements yield a very accurate measurement of the excited-state (6P1/2) hyperfine splitting in Cs, and reveals a breakdown in the well-known Voigt spectral profile. We develop a theoretical model that accounts for this breakdown, explaining the observations to within the shot-noise limit. Our model enables us to infer the thermal velocity dispersion of the Cs vapour with an uncertainty of 35 p.p.m. within an hour. This allows us to determine a value for Boltzmann's constant with a precision of 6 p.p.m., and an uncertainty of 71 p.p.m.

S pectroscopy is a vital tool for both fundamental and applied studies. Improving both the precision and accuracy of spectroscopic measurements is critical to efforts targeted at gaining maximum physical information about the quantum absorbers-even as they are perturbed by the probe. The immediate motivation for our work has arisen out of a call to the scientific community to develop new techniques to remeasure Boltzmann's constant, k B , in preparation for a redefinition of the kelvin 1 ; however, advances in absorption lineshape measurement and theory will find applications in accurate gas detection and monitoring 2 , studies of planetary atmospheres 3 , thermometry in tokamaks 4 and understanding distant astrophysical processes 5,6 . The accurate measurement of natural linewidths and transition frequencies, which can be directly related to the atomic lifetime, level structure and transition probabilities, is important for testing atomic physics [7][8][9] .
At thermal equilibrium, the velocity distribution of atomic absorbers in a vapour cell is related to the temperature through the Boltzmann distribution. This simple and fundamental relationship forms an excellent foundation for a type of primary thermometry known as Doppler broadening thermometry (DBT) 10,11 . DBT differs from the current leading technique for primary thermometry, which measures the temperaturedependent speed of sound in a noble gas contained in a well-characterized acoustic resonator. Using that approach, a breakthrough determination of k B was made in 1988 with a total uncertainty of 1.7 p.p.m. (ref. 12). Refinements to this technique over the last 20 years have reduced its uncertainty to the 1 p.p.m. level 13,14 , improving the uncertainty in the CODATArecommended value for k B from 1.8 p.p.m. (CODATA-02) to 0.91 p.p.m. (CODATA-10). Despite these superb measurements, it is important that different techniques are employed to measure k B to reduce the possibility of underestimating systematic uncertainties that may be inherent to any single technique 15 .
The DBT approach is one of the most theoretically transparent of the new approaches to primary thermometry. The spectrum of a gas sample is measured with high precision, and a theoretical model is fitted to the measured data. If the model includes all of the relevant physics then one can accurately extract the contribution to spectral broadening arising solely from the thermal distribution of atomic/molecular speeds. Ammonia probed by a frequency-stabilized CO 2 laser at 1.34 mm was the first thermometric substance employed in a DBT experiment 16 . Subsequently, an extended-cavity diode laser at 2 mm was used to probe a ro-vibrational transition of CO 2 (ref. 17). In these first experiments, the lineshape was assumed to be a Gaussian or a Voigt profile (a Gaussian convolved with a Lorentzian). Since 2007, with the ambitious goal of approaching 1 p.p.m. accuracy, substantial experimental and theoretical improvements have been made to DBT using ammonia 18,19 , oxygen 20 , ethyne 21 and water 22 . However, one key challenge peculiar to molecular absorbers is the need to account for complex collisional effects on the lineshape 19 . We avoid this by using a dilute atomic vapour 23,24 with a strong dipole transition for which a tractable, microscopic theory has been developed 25,26 . The benefits of a range of different atomic species have been considered in ref. 27.
Here we present measurements of a transmission lineshape of cesium (Cs) vapour with a quantum-limited transmission uncertainty of 2 p.p.m. in a 1 s measurement, which is to our knowledge, a factor of 16 times superior to anything previously demonstrated 28 . This extreme precision, allows us to directly detect subtle lineshape perturbations that have not been previously observed. This observation prompted the development of a theoretical model, which now allows us to discriminate between the internal atomic state dynamics and their external motional degrees. Using the model we are able to estimate the velocity dispersion of the atoms with a precision (s.e.) of 53 p.p.m. during a single-line scan, taking B30 s. This is consistent with the sample s.d. (also 53 p.p.m.) over multiple scans, demonstrating the excellent reproducibility of the spectrometer. The measurement precision averages down to 3.7 p.p.m. after 200 scans. These values are B7 times better than the previous best results [28][29][30] , and also yields a near-two-fold reduction in the uncertainty of the excited-state hyperfine splitting in Cs 31 . Our measurement of the homogenous broadening component of the lineshape has a precision within a factor of two of the best ever measurements of natural linewidth. Modest improvements in the probe laser performance would deliver a new result for the excited state lifetime of Cs in a system that is experimentally and theoretically much simpler than that typically used for lifetime studies 32,33 .

Results
Transmission spectra. We measure the transmission through the vapour cell as the probe laser frequency scans across the Cs D1 transition (6S 1/2 -6P 1/2 ), shown in Fig. 1a. The relative noise in the measurement of the atomic transmission is set by the shot noise in the probe and reference beams, and provides 2 p.p.m. in a 1 s measurement (at the highest optical powers used). The two absorption dips seen in Fig. 1a come from the hyperfine splitting in the excited state and the excellent signal-to-noise ratio enables us to determine this splitting to be f HFS ¼ 1167.716(3) MHz. The accuracy of this measurement exceeds that of the previous best determination 31 in which a sub-Doppler atomic beam spectroscopy technique was used to measure f HFS . Spectroscopy in vapour cells is appealing for its ease of implementation, but as Gerginov et al. identified, such approaches often suffer from optical-pumping-related systematics that necessarily arise in the creation of sub-Doppler absorption features. Here, we avoid this by greatly improving the detection sensitivity and perturbatively accounting for non-linear absorption effects in our lineshape model. The conventional description of an absorption lineshape, T at f ð Þ, of an atomic transition with a rest-frame transition frequency f 0 , is given by a convolution of a Lorentzian function with a natural (half) linewidth G L ¼ 1= 4pt ð Þ due to the finite lifetime t of the transition, with a Gaussian distribution having a e À 1 -width due to Doppler broadening of atoms of mass m at temperature T. By fitting a lineshape to the measured transmission data, with G fit D as a fitting parameter, we extract the Doppler component, from which we infer the temperature. Systematic errors in G fit D are quantified by converting the independent PRT temperature measurements into an expected Doppler width, G PRT D , using the CODATA values for k B and m Cs , which we compare with G fit D . If the natural lineshape of the transition is a Lorentzian, then the resulting convolution is the well-known Voigt profile, which is commonly used to model dipole resonances. However, our extremely low-noise transmission measurements reveal deviations from the Voigt profile. Some of these deviations are technical in origin for example, instrumental broadening due to the lineshape of the probing laser, residual spontaneous emission from the probe laser, unwanted optical etalons and photodetector linearity; however, there is an important fundamental effect resulting from frequency-dependent optical pumping, which perturbs the atomic natural lineshape away from a Lorentzian. All of these effects, whether technical or fundamental, cause systematic perturbations to the lineshape, and must be taken into account to model the lineshape accurately.
Least-squares fitting. To demonstrate these deviations, we fit a model consisting of two Voigt profiles separated by f HFS , to the raw transmission data shown Fig. 1a. The residuals of this Voigt-only fit have characteristic 'M'-shaped features near resonance, with amplitude B200 p.p.m. at the highest probe powers, as shown in the green trace in Fig. 1b. These features demonstrate the breakdown of the Voigt profile, indicating additional, unaccounted for physics, which causes a spurious linear dependence of G fit D on probe intensity, as shown in the green curve in Fig. 2. In what follows, we describe how we sequentially include additional physics in our transmission model to remove these, and other, systematic effects, leaving only the shot-noise limit of our detection apparatus.
In earlier work we showed that the 'M'-shaped features in Fig. 1b arise from atomic population dynamics induced by the probe laser, which are significant even at exceedingly low intensities 25 . We subsequently calculated corrections to the Voigt profile up to linear order in the probe intensity 26 , to account for optical pumping. Including this first-order intensitydependent corrections in our model suppresses the features substantially, as shown in the blue trace in Fig. 1b.
where f j is the free-spectral range of etalon j.
where n ¼ G D /G L , D is the detuning from the resonance, and the generalized Voigt profile is given by V n ð Þ n D ð Þ ¼ p À 3=2 R 1 À 1 dx e À D À x ð Þ 2 =n 2 1 þ x 2 ð Þ À n , and the conventional Voigt pro- For a cell of length L with linear absorption coefficient a, the optical depth on resonance is p ¼ aL, q 1 pI is the linear intensity-dependent coefficient 25 , and q 2,3 pI 2 are quadratic intensity-dependent coefficients.
The red-dotted trace in Fig. 1d shows the residuals after including second-order intensity corrections. The residuals are suppressed to B2 p.p.m., consistent with the detection noise floor, giving confidence that the transmission model accounts for systematic effects. Figure 2 compares the fractional deviation between G fit D and G PRT D as a function of incident intensity for the three possible models for the atomic lineshape: Voigt-only (green), first-order (blue) and second-order (red) intensity-dependent corrections. Naturally, the true G D should be intensity independent. The simplest Voigt-only theory clearly exhibits a linear intensity dependence, leading to B750 p.p.m. discrepancy even at intensities as low as I/I sat ¼ 2.8 Â 10 À 3 where I sat ¼ 2.5 mW cm À 2 is the saturation intensity for the transition 35 . From ref. 26, we expect the first-order correction to have a residual quadratic dependence on intensity, which is consistent with the blue curve on Fig. 2. Finally, the second-order correction (red curve) is seen to suppress all intensity dependence to below the measurement precision. The simultaneous removal of all systematic features from the residuals together with the elimination of intensity dependence in the recovered G fit D gives a high degree of confidence that the model includes all relevant physics.
We note briefly that in fluorescence spectroscopy one should account for quantum interference between different hyperfine manifolds [36][37][38][39][40] . This is not an issue in absorption spectroscopy (see the Quantum Interference section of the Supplementary Discussion).
Application to primary thermometry. In this section we demonstrate the power of our spectroscopic technique by applying it to the problem of primary thermometry, and determine a value for Boltzmann's constant using atomic spectroscopy.
The atomic vapour is thermalized at around T ¼ 296 K (that is, G PRT D ¼ 215.123 MHz) by suspending it in a thermal isolator. The independent PRT measurement has an uncertainty of 0.55 mK (1.9 p.p.m.). We note the temperature drifted over the course of the experimental scans, such that G PRT D is expected to vary from B214.96 to 215.13 MHz (see Supplementary Fig. 7).
After systematically removing the effects of optical pumping and etalons, the largest source of uncertainty in G fit D comes from the uncertainty in G L . We obtain a value for G L by directly fitting it to the data, which gives G fit L ¼ 2.327 (7) MHz. This is consistent with, but also more precise than, an independent estimate, G ind L ¼ 2.331 (19) MHz, given by the sum of the natural linewidth of the Cs 6P 1/2 level, G at L ¼ 2.287(6) MHz (ref. 35), and the laser linewidth, G las L ¼ 0.044(18) MHz. From Fig. 2 we estimate that a 1 kHz change in G L leads to B5 p.p.m. change in G fit D , so that the 6.5 kHz uncertainty in G L contributes 32.5 p.p.m. uncertainty in G fit D (65 p.p.m. in u r (k B )). We note that if the probe linewidth were B1 kHz, then our measurement would yield a state-of-the-art value for the excited-state lifetime of Cs. We now briefly describe additional sources of uncertainty, which are quantified in Table 1 (see the Error Budget section of the Supplementary Discussion for a comprehensive description). Statistical error arising from the scatter in Fig. 3 contributes 2.9 p.p.m. to u r G fit D À Á (5.8 p.p.m. to u r (k B )). This is also consistent Optical pumping shifts are reported at I/I sat ¼ 3 Â 10 À 3 assuming second-order corrections to the Voigt profile. u r denotes fractional uncertainty. *Estimated from fits to scan data with free Lorentz width. w Estimated from published uncertainty in Cs lifetime, plus uncertainty in the independent estimate of the laser linewidth.  To determine a value for k B , we take a weighted mean of G fit D =G PRT D extracted from fits using all the second-order corrected data shown on Fig. 2 (red points). We find k B ¼ 1.380545(98) Â 10 À 23 J K À 1 , where the 71 p.p.m. uncertainty is calculated in Table 1. This is consistent with the recommended CODATA value of 1.3806488(13) Â 10 À 23 J K À 1 .

Discussion
In conclusion, we have developed an atomic absorption spectrometer that operates with a transmission measurement precision of just 2 p.p.m. This has revealed several new phenomena. Combined with a perturbative treatment of optical pumping we are able to explain all of the observed effects to a level of precision never before demonstrated. The power of our technique is demonstrated by measuring the lineshape parameters of thermal vapour. The reproducibility is consistent with the quantum limits of measurement 41 giving confidence that we have captured the relevant physics.
With this unprecedented sensitivity, we have measured the transmission spectrum of Cs at 895 nm at the laser intensity shotnoise limit. From these spectra, we derive a value for Boltzmann's constant with an uncertainty of 71 p.p.m., consistent with the recommended CODATA value, and the hyperfine frequency splitting of the 6 2 P 1/2 level with the lowest uncertainty ever previously reported: Although drifts in the experiment are likely in the apparatus on timescales required for multiple scans (for example, in the absolute optical frequency of the master laser), we average only the fitted parameters from each spectrum that are insensitive to the drifts; for example, the centre frequency of each resonance cannot be determined to a better accuracy than the stability of the master laser, but their difference can be averaged over long timescales to achieve a lower uncertainty.
A future experiment designed to further minimize etalons, and using a higher power, sub-kHz linewidth probe laser with greater tuning range (B30 GHz) would eliminate probe laser effects, allow better estimation of etalon contamination, and lower the contribution of optical pumping. This opens the door to the best measurement of the Cs D1 transition lifetime, and a p.p.m.-level measurement of k B thus making an important contribution to efforts to redefine the kelvin.

Methods
Probe light preparation. The spectrometer is pictured in Fig. 4a. The probe laser (P) is tightly locked at a tunable frequency difference away from a master laser (M), which is itself stabilized to an atomic transition in Cs. The frequency stability of the master laser is compared with a reference frequency provided by an optical frequency comb (C). The probe laser is spectrally and spatially filtered using a combination of an optical cavity (OC) of moderate finesse F % 305 ð Þand singlemode fibre (not shown). This reduces the spontaneous emission content of the probe beam from 1.6% to below 0.01%. The optical power is actively stabilized with an acousto-optic modulator (AOM). It is then delivered into a vacuum chamber in which the vapour cell is mounted in a thermal and magnetic shield. Calibrated standard platinum resistance thermometers (SPRTs) were used to independently measure the temperature of the shield and control it with a precision of a few millikelvin. Spatial temperature gradients are suppressed to the same level.
Measurement scheme. Inside the vacuum chamber, the probe light is split into two beams using a combination of a Glan-Taylor polarizing prism (GT) and Wollaston beam splitter (W). One beam passes through a Cs cell embedded inside the thermal and magnetic shield and is measured by photodiode A (PD A); the other beam is measured directly by photodiode B (PD B). The ratio of these photodiode signals gives us the transmission ratio, T , which was stable to better than 10 À 6 . The incident power is actively controlled using the level on PD A while the frequency of the incident light is set with an absolute uncertainty of 2 kHz, limited by our ability to realize the unperturbed transition frequency of the Cs saturated absorption feature to which the master laser was locked. Precision in the radio frequency (RF) frequency detuning between the master and probe lasers was ensured by referencing the offset-frequency locking electronics to a Cs beam clockdisciplined maser. Technical details about the apparatus are given in the Supplementary Methods section. Cs Cell Figure 4 | High-accuracy linear absorption spectrometer for probing the D1 transitions in atomic Cs vapour. A probe laser (P) is offset-frequency locked to a master laser (M), which is in turn compared against a stabilized optical frequency comb (C). The probe laser output is spectrally and spatially filtered with an optical cavity (OC) before being deflected by an acouto-optic modulator (AOM). The beam is then passed through a Glan-Taylor (GT) prism and divided on a Wollaston (W) beam splitter. The power of the beam transmitted through the Cs cell inside the thermal and magnetic shield is measured on a photodetector PDA and compared with a reference beam measured on photodetector PDB. SPRT1 and SPRT2 are standard platinum resistance thermometers from which we independently measure the cell temperature.